Question

The freezing-point depression of a \(0.091-m\) solution of \(\mathrm{CsCl}\) is \(0.320^{\circ} \mathrm{C} .\) The freezing-point depression of a \(0.091-\mathrm{m}\) solution of \(\mathrm{CaCl}_{2}\) is \(0.440^{\circ} \mathrm{C} .\) In which solution does ion association appear to be greater? Explain.

Short answer

The greater ion association appears to be in the CsCl solution. This is because the experimental Van't Hoff factor (i) for CsCl is lower than that for CaCl₂, indicating a higher degree of ion association in the CsCl solution. The experimental Van't Hoff factors were calculated using the formula: i = (observed freezing-point depression) / (theoretical freezing-point depression), and the values found were 0.320°C / 2 for CsCl and 0.440°C / 3 for CaCl₂. Since the experimental Van't Hoff factor for the CsCl solution is lower, it has greater ion association.

Step-by-step solution

Calculate the observed freezing-point depression for each solution

We are given the values for the observed freezing-point depression for both solutions: ΔTf_CsCl = 0.320°C ΔTf_CaCl₂ = 0.440°C

Calculate the theoretical freezing-point depression for each solution

For CsCl, there is one cation (Cs⁺) and one anion (Cl⁻) per formula unit, so the total number of ions is 2. Therefore, the theoretical van't Hoff factor for CsCl is i_CsCl = 2. For CaCl₂, there is one cation (Ca²⁺) and two anions (2 Cl⁻) per formula unit, so the total number of ions is 3. Therefore, the theoretical van't Hoff factor for CaCl₂ is i_CaCl₂ = 3. Now, we can use the formula ΔTf = Kf * molality * i to calculate the theoretical freezing-point depression for each solution: ΔTf_CsCl = Kf * (0.091 m) * i_CsCl ΔTf_CaCl₂ = Kf * (0.091 m) * i_CaCl₂

Calculate the experimental Van't Hoff factors for each solution

We can now calculate the experimental Van't Hoff factors for each solution using the formula mentioned in the analysis: i_CsCl = (observed freezing-point depression) / (theoretical freezing-point depression) i_CsCl = 0.320°C / (Kf * (0.091 m) * 2) i_CaCl₂ = 0.440°C / (Kf * (0.091 m) * 3)

Compare the experimental van't Hoff factors to determine the ion association

The solution with the lower experimental Van't Hoff factor will have a greater degree of ion association since the expected freezing-point depression will not be achieved. We can see that i_CsCl and i_CaCl₂ are both divided by the same Kf and molality (0.091 m), so by comparing their observed freezing-point depression values (0.320°C for CsCl and 0.440°C for CaCl₂), we can conclude that the solution with the greater ion association is the one with the lower observed freezing-point depression: 0.320°C / 2 < 0.440°C / 3 Since 0.320°C / 2 (CsCl) is less than 0.440°C / 3 (CaCl₂), the greater ion association is observed in the CsCl solution.

Key concepts

van't Hoff factor

The van't Hoff factor, often denoted as "i," is a critical concept in understanding colligative properties such as freezing-point depression. It's a measure of the degree of ionization or dissociation a solute undergoes in solution. When a solute like an ionic compound dissolves in a solvent, it typically separates into its respective ions.

For instance, the compound CsCl splits into Cs⁺ and Cl⁻ ions. Therefore, in the case of CsCl, the expected van't Hoff factor would be 2, indicating each formula unit breaks into two particles. Similarly, CaCl₂ dissociates into one Ca²⁺ and two Cl⁻ ions, resulting in a theoretical van't Hoff factor of 3.

It's important to note that the van't Hoff factor alters the freezing-point depression calculation by multiplying the original molality with the factor. Thus, a higher van't Hoff factor indicates more particles in the solution, which leads to a greater change in the freezing point. However, ion association can complicate this prediction, as we'll explore further.

ion association

Ion association is when ions in a solution are not completely dissociated but instead, some of them remain paired. This affects the effective number of particles contributing to colligative properties like freezing-point depression.

When you calculate the theoretical van't Hoff factor based on complete dissociation, you assume all ionic bonds break. However, in reality, some ions may still interact, reducing the number of distinct species in the solution. This creates a scenario where the experimental van't Hoff factor is less than expected.

• Higher ion association implies fewer free particles in solution, leading to a lower experimental van't Hoff factor.
• This results in a smaller observed effect on freezing-point depression compared to theoretical predictions.

Thus, a significant ion association indicates strong interactions between ions, resulting in a less effective contribution to changing the solution's freezing point.

molality

Molality is a concentration term used in chemistry particularly involving colligative properties. It is defined as the number of moles of solute per kilogram of solvent, and is represented as "m."

Molality plays a crucial role because it is independent of temperature and volume changes, making it a preferred measurement in colligative property calculations. In the context of the freezing-point depression, the change in temperature, \( \Delta T_f \), is directly proportional to the molality of the solution. The formula used is:\[\Delta T_f = K_f \times \text{molality} \times i\]

Here, \( K_f \) is the cryoscopic constant (a property of the solvent), and \( i \) is the van't Hoff factor. Thus, by knowing a solution's molality, you can accurately predict its expected change in freezing point, provided the ideal dissociation scenario (with the van't Hoff factor) holds true.